There are several sets of axioms for both Euclidean and hyperbolic plane geometry. If the axioms in each such set are combined, we could say that we have definitions of Euclidean or hyperbolic planes. My goal is to describe one formulation and how the uniqueness theorems can be proved in that context. In this formulation, the real numbers are taken as a starting point, rather than deriving the real numbers from other properties of planes. The two uniqueness theorems have a number of steps in common, consisting of what are called neutral theorems, i.e., ones that do not depend on any assumption about the number of lines parallel to a given line through a given point. Indeed one of the theorems is that there is always at least one such parallel line. I will only mention some of the theorems more central to the development.

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